Trigonometry: A day at the track

Let's suppose we are watching a foot race and while watching the runners, our minds turn in a mathematical direction. Let's focus on one of the runners as she runs in a counter-clockwise direction about the circular track. She is wearing a blue hat as she runs at a constant speed. The picture below shows what we might see from three vantage points.

We have three observers who can record her position. One is a photographer directly overhead, in the sponsor's blimp, and two others are on the ground in the bleachers at the East (E) and South (S) end of the track. The photographer in the blimp sees the picture shown above. However, the two observers on the ground see her running past, first one way and then the other. Each one of them can only see a one-dimensional view of her motion, and it might look as though she is bouncing back and forth, like a ping-pong ball.
You might think that what these two ground observers are seeing is quite similar. Indeed, in many ways the picture that they see is the same, but they see it at different times. This is what we call a phase difference between one view and the other. In this case, the phase difference is due to the fact that it takes the runner 1/4 revolution to get from (S) to (E).
Describing what these two observers see leads to some very interesting mathematical functions. In fact, this type of situation---where we see only the projection of a more complicated motion---is very common in the world around us. Our efforts here will give us some powerful tools to study natural phenomena.

Sines and Cosines

We can enlist the help of a coordinate system to describe this situation. Suppose that the track is the unit circle and that the runner starts at the point (1,0). After running a distance t, she will arrive at another point on the unit circle with some x and y coordinates. We call this point $(\cos(t), \sin(t))$ .
The following picutre demonstrates this:

From this picture, all of the mysteries of the sine and cosine functions are revealed. In particular, all of the basic relationships between them can be understood in terms of this picture.

Properties of Sines and Cosines

An excellent way to acquaint yourself with these functions is to explain how the following properties arise from the geometry of the unit circle.

$\cos^2(t) + \sin^2(t) = 1$ . This basic identity follows from the fact that $(\cos(t), \sin(t))$ always lies on the unit circle.
Periodicity: $\begin{eqnarray*} \cos(t+2\pi) & = & \cos(t) \\ \sin(t+2\pi) & = & \sin(t) \end{eqnarray*}$
This property is perhaps the most important: it is saying that the behavior of these trigonometric functions is periodic . In other words, the functions repeat their behavior every so often (in this case, after an interval of $2\pi$ ). If we look at the world around us, we see many kinds of periodic behaviour: the sun rises every 24 hours, the tides go up and down and our hearts beat once a second. These functions will be vitally important in describing these kinds of phenomena.
Graphs: We can understand the graph of the sine function from the following demonstration. Of course, we have only shown one cycle of the graph but in light of the periodicity property above, it is enough to understand just one cycle.

We can understand the cosine function in the same way:

Symmetry: $\begin{eqnarray*} \cos(-t) & = & \cos(t) \\ \sin(-t) & = & -\sin(t) \end{eqnarray*}$
Reflection in the line y = x: $\begin{eqnarray*} \cos(t) & = & \sin(\frac \pi 2 - t) \\ \sin(t) & = & \cos(\frac \pi 2 - t) \end{eqnarray*}$
Notice that this second relationship, combined with the symmetry property of the cosine function implies that $\sin(t) = \cos(t - \frac \pi 2)$ which expresses the phase difference we noted above.
Sums of angles: $\begin{eqnarray*} \cos(A+B) & = & \cos(A)\cos(B) - \sin(A)\sin(B) \\ \sin(A+B) & = & \sin(A)\cos(B) + \cos(B)\sin(A) \end{eqnarray*}$
These require a little more work to see from the geometry of the circle. However, they will be extremely important for us when we study differential equations.
Double angles: $\begin{eqnarray*} \cos(2A) & = & \cos^2(A) \sin^2(A) \\ \sin(2A) & = & 2\sin(A)\cos(A) \end{eqnarray*}$
Half angles: $\begin{eqnarray*} \cos(\frac A2) & = & \cos^2(A) \sin^2(A) \\ \sin(2A) & = & 2\sin(A)\cos(A) \end{eqnarray*}$
Special Values:

degrees radians sin cos

0 0 0 1

30 $\pi/6$ $\sqrt{3}/2$

45 $\pi/4$ $1/\sqrt{2}$ $1/\sqrt{2}$

60 $\pi/3$ $\sqrt{3}/2$

90 $\pi/2$

Relations with High-School Trigonometry

You have probably seen these functions before in high school though perhaps within the context of right triangles. The following demonstration aims to clarify the relationship between these two approaches.

Remember that the arclength, which we have been calling t, around a circle of radius r is related to the central angle $\theta$ as
$t = r\theta$
In our case, since the radius r = 1, we have that $t = \theta$ . The demonstration above shows two similar right triangles: one has legs whose lengths are $\cos(\theta)$ and $\sin(\theta)$ and hyptonuse 1. By the similarity of the triangles, we see that
$\cos(\theta) = \frac{\cos(\theta)}{1} = \frac{\mbox{adj}}{\mbox{hyp}}$
where adj denotes the length of the leg adjacent to the angle and hyp the length of the hyptonuse. In the same way, if opp is the length of the opposite leg,
$\sin(\theta) = \frac{\mbox{opp}}{\mbox{hyp}}$
This shows that these functions record the appropriate ratios of the legs of right triangles and in fact, trigonometric functions are important for this way of representing angles. However, this situation is generally considered to be static and used mainly by, say, navigators who need to get a precise fix on the distance to some faraway land or carpenters who need to get an accurate structure built. Thalus, one of the Greek explorers, used a process akin to trigonometry to estimate the size of the Great Pyramid of Cheops.
However, these aspects of the trig functions, though important in their own right, are rather mundane and tame. Their true nature is much more fundamental and important in scientific applications: sines and cosines are the ideal way to represent cyclical or periodic phenomena. And there are lots of such phenomena in nature: the change of the seasons, the cycles of day and night, the backwards and forward swing of a pendulum, the up and down motion of waves on a beach, the vibrations of a tuning fork and a speaker that produces audible oscilations (sound waves). Nature is full of such processes, and in one way or another, sines and cosines are always used by scientists to represent such things.
Just for the record, you are probably familiar with these other trigonometric functions as well:

$\tan(t) = \frac{\sin(t)}{\cos(t)} \hspace{.75in} \sec(t) = \frac{1}{\cos(t)}$

degrees	radians	sin	cos
0	0	0	1
30	$\pi/6$		$\sqrt{3}/2$
45	$\pi/4$	$1/\sqrt{2}$	$1/\sqrt{2}$
60	$\pi/3$	$\sqrt{3}/2$
90	$\pi/2$